a Calculate the relative frequency of sending 1500 or fewer messages. To do so divide the frequency of sending 1500 or fewer messages by the total of the frequencies.
B
b Calculate the relative frequency of sending more than 1500 messages. To do so divide the frequency of sending more than 1500 messages by the total of the frequencies.
A
a 145/983
B
b 838/983
Practice makes perfect
a We are given a table showing the numbers of text messages sent in one month by students at Metro High School. First we want to find the probability that a student chosen at random sends 1500 or fewer text messages in one month. Consider the given table.
Number of Text Messages t
Number of Students
t ≤ 500
25
500
120
1500
300
t>2500
538
To find the desired probability we need to calculate the relative frequency. To do so we need to divide the frequency of sending 1500 or fewer messages by the total of the frequencies. First, let's calculate the total of the frequencies. To do so, we need to add all the frequencies together.
Total: 25+120+300+538= 983
Let's now find the frequency of sending 1500 or fewer messages. Consider our table once again to see which frequencies match this description.
Number of Text Messages t
Number of Students
t ≤ 500
25
500
120
1500
300
t>2500
538
Since less than or equal to 500 is still fewer than 1500, we have two frequencies satisfying our condition. To find the frequency of sending 1500 messages or fewer, we need to add them.
1500 or fewer: 25+120= 145
Finally, we can calculate the desired relative frequency by dividing the two obtained frequencies.
Relative frequency= 145/983
Therefore, a probability of choosing a student who sends 1500 messages or fewer is 145983.
b Now, we want to find the probability that a student chosen at random sends more than 1500 messages in a month. Consider the given table.
Number of Text Messages t
Number of Students
t ≤ 500
25
500
120
1500
300
t>2500
538
To find the desired probability we need to calculate the relative frequency of sending more than 1500 messages. To do so we need to divide the frequency of sending more than 1500 messages by the total of the frequencies. The total of the frequencies is the same as before.
Total: 983
Let's now find the frequency of sending more than 1500 messages. Consider our table once again to see which frequencies match this description.
Number of Text Messages t
Number of Students
t ≤ 500
25
500
120
1500
300
t>2500
538
Since more than 2500 is still more than 1500, we have two frequencies satisfying our condition. To find the frequency of sending more than 1500 messages we need to add them.
More than1500: 300+538= 838
Finally, we can calculate the desired relative frequency by dividing this frequency by the total of the frequencies.
Relative frequency= 838/983
Therefore, the probability of choosing a student who sends more than 1500 messages is 838983.
Alternative Solution
Alternative Way of Finding the Probability
Let's now consider another way to solve this part of the exercise. For the alternative solution it is crucial to notice that the event we are interested in is a complement of the event of sending 1500 or fewer messages. Recall how we calculate the probability of a complement.
P(not event)=1-P(event)
In the first part of the exercise we found that the probability of choosing a student who sends 1500 or fewer messages is 145983. Let's use this fact and the formula for the probability of a complement to find the probability of picking a student who sends more than 1500 messages a month.
